The mathematics of the discrete Fourier transform

We aim to identify the assumptions that are implicit in the sampling of a continuous-time signal and in the subsequent application of the discrete Fourier transform (DFT). In particular, we consider the following questions:

When does the sampling of periodic continuous-time signal result in a periodic discrete-time signal?

When the resulting discrete-time signal is periodic, what is its frequency in samples/second?

Which continuous-time frequencies coincide in discrete time, and what does the “frequency spectrum” in discrete-time look like?

To which periodic discrete-time signals can the discrete Fourier transform be applied to without losing information?

We furthermore show that the DFT interchanges point-wise and convolution products in the time- and frequency- domains, and thereby express the DFT to Pontryagin duality for finite cycle groups.